A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance

Marco Martens; Vincent Naudot; Jiazhong Yang

doi:10.1016/j.crma.2005.04.001

Dynamical Systems/Ordinary Differential Equations

A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance

Presented by: Étienne Ghys

Marco Martens ¹; Vincent Naudot ¹; Jiazhong Yang ²

¹ University of Groningen, Department of Mathematics, P.O. Box 800m, NL-9700 AV Groningen, The Netherlands
² LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China

Comptes Rendus. Mathématique, Volume 340 (2005) no. 11, pp. 843-846.

Abstract
Résumé

In this Note, we study the unfolding of a vector field that possesses a degenerate homoclinic (of inclination-flip type) to a hyperbolic equilibrium point where its linear part possesses a resonance. For the unperturbed system, the resonant term associated with the resonance vanishes. After suitable rescaling, the Poincaré return map is a cubic Hénon-like map. We deduce the existence of a strange attractor which persists in the Lebesgue measure sense. We also show the presence of an attractor with topological entropy close to $\log 3$ .

Nous étudions le déploiement d'un champ de vecteurs sur $R^{3}$ qui possède une orbite homocline dégénérée associée à une singularité hyperbolique. La partie linéaire du champ en cette singulartité posssède une résonance mais, pour le système initial, le terme résonant associé à cette résonance disparaît. Nous montrons qu'après changement d'échelle, l'application de retour de Poincaré sur une section transverse est proche d' une application de Hénon cubique. Un attracteur étrange est présent et persiste au sens de la mesure de Lebesgue. Nous montrons également la présence d'un attracteur avec une entropie topologique proche de $\log 3$ .

Received: 2005-03-02
Accepted: 2005-03-27
Published online: 2005-05-23

DOI: 10.1016/j.crma.2005.04.001

Author's affiliations:

Marco Martens ¹; Vincent Naudot ¹; Jiazhong Yang ²

¹ University of Groningen, Department of Mathematics, P.O. Box 800m, NL-9700 AV Groningen, The Netherlands
² LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China

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     author = {Marco Martens and Vincent Naudot and Jiazhong Yang},
     title = {A cubic {H\'enon-like} map in the unfolding of degenerate homoclinic orbit with resonance},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {843--846},
     publisher = {Elsevier},
     volume = {340},
     number = {11},
     year = {2005},
     doi = {10.1016/j.crma.2005.04.001},
     language = {en},
}

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Marco Martens; Vincent Naudot; Jiazhong Yang. A cubic Hénon-like map in the unfolding of degenerate homoclinic orbit with resonance. Comptes Rendus. Mathématique, Volume 340 (2005) no. 11, pp. 843-846. doi : 10.1016/j.crma.2005.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.04.001/

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